Gradient Ricci solitons with half harmonic Weyl curvature and two Ricci eigenvalues
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In this article we classify four dimensional gradient Ricci solitons $(M, g, f)$ with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, $(V, g)$ is isometric to one of the following: {\rm (i)} an Einstein manifold. {\rm (ii)} a domain in the Riemannian product $ (\mathbb{R}^2, g_0) \times (N, \tilde{g})$, where $g_0$ is the flat metric on $\mathbb{R}^2$ and $(N, \tilde{g})$ is a two dimensional Riemannian manifold of constant curvature $\lambda \neq 0$. {\rm (iii)} a domain in $\mathbb{R} \times W$ with the warped product metric $ ds^2 + h(s)^2 \tilde{g},$ where $\tilde{g}$ is a constant curved metric on a three dimensional manifold $W$.
Abstract
In this article we classify four dimensional gradient Ricci solitons $(M, g, f)$ with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, $(V, g)$ is isometric to one of the following: {\rm (i)} an Einstein manifold. {\rm (ii)} a domain in the Riemannian product $ (\mathbb{R}^2, g_0) \times (N, \tilde{g})$, where $g_0$ is the flat metric on $\mathbb{R}^2$ and $(N, \tilde{g})$ is a two dimensional Riemannian manifold of constant curvature $\lambda \neq 0$. {\rm (iii)} a domain in $\mathbb{R} \times W$ with the warped product metric $ ds^2 + h(s)^2 \tilde{g},$ where $\tilde{g}$ is a constant curved metric on a three dimensional manifold $W$.
참고문헌 (8)
- A. L. Besse, "Einstein manifolds" (1987), Springer-Verlag
- Huai-Dong Cao, "On locally conformally flat gradient steady Ricci solitons", Transactions of the American Mathematical Society, 364(5), pp.2377~2391 (2012), American Mathematical Society (AMS)[DOI]
- Huai-Dong Cao, "On Bach-flat gradient shrinking Ricci solitons", Duke Mathematical Journal, 162(6), pp.1149~1169 (2013), Duke University Press[DOI]
- Xiuxiong Chen, "On Four-Dimensional Anti-self-dual Gradient Ricci Solitons", The Journal of Geometric Analysis, 25(2), pp.1335~1343 (2014), Springer Science and Business Media LLC[DOI]
- B. Chow, "The Ricci flow: techniques and applications. Part I" (2007), American Mathematical Society
- Andrzej Derdziński, "Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor", Mathematische Zeitschrift, 172(3), pp.273~280 (1980), Springer Science and Business Media LLC[DOI]
- Jongsu Kim, "On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature", The Journal of Geometric Analysis, 27(2), pp.986~1012 (2016), Springer Science and Business Media LLC[DOI]
- Jia-Yong Wu, "Gradient shrinking Ricci solitons of half harmonic Weyl curvature", Calculus of Variations and Partial Differential Equations, 57(5), pp.15~ (2018), Springer Science and Business Media LLC[DOI]